Non-holonomic stability and rotation with zero angular momentum: Demonstrations of driving stability and of the falling cat phenomenon go sour
There are two classes of interesting, at least to me, physical behavior that follow from the impossibility of integrating some formulas that involve derivatives. I learned of both of them from Tom Kane. First, systems with wheels or ice skates can conserve energy yet still act damped. This occurs despite the supposed theorem that conservative systems cannot have such stability. In fact (energy conserving) cars, flying arrows, skateboards and bicycles can have this stability. Second is the well-known possibility that a system with zero angular momentum can, by appropriate deformations, rotate without any external torque. It is not rotating, it has zero angular momentum, yet it gets rotated. This is the falling cat problem: a cat dropped upside down can turn over.
Both rolling contact and constancy of angular momentum are examples of "non-integrability" of a "non-holonomic" equation. There are various simple demonstrations of these phenomena that can go bad. Cars can crash, bikes can fall over. And, in terrestrial angular-momentum experiments, various sometimes-subtle effects can swamp that which one wants to demonstrate. The talk describes the basic theory and also various simple experiments that fail various ways for various reasons.
With luck, the talk will be somewhat accessible to perky undergraduates, while still being somewhat amusing to those more technically savvy.